I am interested in determining a collection of geometric conditions that will guarantee that a convex polyhedron
of $n$ faces is a _fair die_ in the sense that, upon random rolling, it has an equal $1/n$ probability of
landing on each of its faces.
(Assume the polyhedron is composed of a homogeneous material; i.e., it is not "loaded.")
There has been study of what Grünbaum and Shephard call _isohedral_ polyhedra,
which always represent fair dice: "An [isohedron][1] is a convex polyhedron with symmetries acting transitively on its faces with respect to the center of gravity. Every isohedron has an even number of faces."
It is clear such a polyhedral die is fair.
Here is an example of the _trapezoidal dodecahedron_, an isohedron of 12 congruent faces,
from [an attractive web site on polyhedral dice][2]:
<br />
![trapezoidal dodecahedron][6]
<br />
But a clever argument in a delightful paper
by Persi Diaconis and Joseph Keller
("[Fair Dice][4]." _Amer. Math. Monthly_ 96, 337-339, 1989)
shows (essentially, by continuity) that there must be fair polyhedral dice
that are not symmetric.
For example, there is no reason to expect that equal face areas is a necessary condition
for a polyhedral die to be fair.
Nor is it reasonable to expect that the distance from each face to the center of gravity
of the polyhedron is alone a determining condition.
Rather it should depend on the dihedral angles between faces, the likelihood of one face
rolling to the next—perhaps a [Markov chain][5] of transitions?
My question is:
> Is there a collection of geometric conditions—broader than isohedral—that guarantee
that a (perhaps asymetrical, perhaps unequal-face-areas) convex polyhedron represents
a fair die?
Sufficient conditions welcomed; necessary and sufficient conditions may be too much to hope for!
Speculations and literature leads appreciated!
[1]: http://mathworld.wolfram.com/Isohedron.html
[2]: http://www.aleakybos.ch/Shapes.htm
[3]: https://people.csail.mit.edu/~orourke/MathOverflow/TrapDodeca.jpg
[4]: http://www-stat.stanford.edu/~cgates/PERSI/papers/fairdice.pdf
[5]: http://en.wikipedia.org/wiki/Markov_chain
[6]: https://i.sstatic.net/uEZdf.jpg